There are $2^{10}=1024$ possible 10-letter strings in which each letter is either an A or a B. Find the number of such strings that do not have more than 3 adjacent letters that are identical.

Each diagonal of the base and each lateral edge of a $9$-gonal pyramid is colored either green or red. Show that there must exist a triangle with the vertices at vertices of the pyramid having all three sides of the same color.

A $\text{0.3 kg}$ apple falls from rest through a height of $\text{40 cm}$ onto a flat surface. Upon impact, the apple comes to rest in $\text{0.1 s}$, and $\text{4 cm}^2$ of the apple comes into contact with the surface during the impact. What is the average pressure exerted on the apple during the impact? Ignore air resistance. (A) $\text{67,000 Pa}$ (B) $\text{21,000 Pa}$ (C) $\text{6,700 Pa}$ (D) $\text{210 Pa}$ (E) $\text{67 Pa}$

In a convex pentagon, let the perpendicular line from a vertex to the opposite side be called an altitude. Prove that if four of the altitudes are concurrent at a point then the fifth altitude also passes through this point.

Prove that there is no natural number n such that the sum of all the digits of the number m, where $m = n(2n-1)$ is equal to $2000$.

solve $x^2+31=y^3$ in integers

Consider $2009$ cards, each having one gold side and one black side, lying on parallel on a long table. Initially all cards show their gold sides. Two player, standing by the same long side of the table, play a game with alternating moves. Each move consists of choosing a block of $50$ consecutive cards, the leftmost of which is showing gold, and turning them all over, so those which showed gold now show black and vice versa. The last player who can make a legal move wins. (a) Does the game necessarily end? (b) Does there exist a winning strategy for the starting player? [i]Proposed by Michael Albert, Richard Guy, New Zealand[/i]

Find all the functions $ f: \mathbb{N}\rightarrow \mathbb{N}$ such that \[ 3f(f(f(n))) \plus{} 2f(f(n)) \plus{} f(n) \equal{} 6n, \quad \forall n\in \mathbb{N}.\]

A $1\times 2019$ board is filled with numbers $1, 2, \dots, 2019$ in an increasing order. In each step, three consecutive tiles are selected, then one of the following operations is performed: $\text{(i)}$ the number in the middle is increased by $2$ and its neighbors are decreased by $1$, or $\text{(ii)}$ the number in the middle is decreased by $2$ and its neighbors are increased by $1$. After several such operations, the board again contains all the numbers $1, 2,\dots, 2019$. Prove that each number is in its original position.

Let $ p_1, p_2, p_3$ and $ p_4$ be four different prime numbers satisying the equations $ 2p_1 \plus{} 3p_2 \plus{} 5p_3 \plus{} 7p_4 \equal{} 162$ $ 11p_1 \plus{} 7p_2 \plus{} 5p_3 \plus{} 4p_4 \equal{} 162$ Find all possible values of the product $ p_1p_2p_3p_4$

It is known that for some integers $a_{2021},a_{2020},...,a_1,a_0$ the expression $$a_{2021}n^{2021}+a_{2020}n^{2020}+...+a_1n+a_0$$ is divisible by $2021$ for any arbitrary integer $n$. Is it required that each of the numbers $a_{2021},a_{2020},...,a_1,a_0$ also divisible by $2021$?

Prove that $[\sqrt{n}+\sqrt{n+1}]=[\sqrt{4n+1}]$ for all $n \in N$.

Prove that \[ \lim_{n \to \infty} n \left( \frac{\pi}{4} - n \int_0^1 \frac{x^n}{1+x^{2n}} \, dx \right) = \int_0^1 f(x) \, dx , \] where $f(x) = \frac{\arctan x}{x}$ if $x \in \left( 0,1 \right]$ and $f(0)=1$. [i]Dorin Andrica, Mihai Piticari[/i]

The incircle of a non-isosceles triangle $ABC$ with the center $I$ touches the sides $BC,AC,AB$ at $A_{1},B_{1},C_{1}$ . let $AI,BI,CI$ meets $BC,AC,AB$ at $A_{2},B_{2},C_{2}$. let $A'$ is a point on $AI$ such that $A_{1}A'\perp B_{2}C_{2}$ .$B',C'$ respectively. prove that two triangle $A'B'C',A_{1}B_{1}C_{1}$ are equal.

On a circumference of a circle, seven points are selected, at which different positive integers are assigned to each of them. Then fit simultaneously, each number is replaced by the least common multiple of the two neighboring numbers to it. If the same number $n$ is obtained in each of the seven points, determine the smallest possible value for $n$. [hide=original wording]Sobre una circunferencia de un círculo, se seleccionan siete puntos, a los cuales se le asignan enteros positivos distintos a cada uno de ellos. Luego, en forma simultánea, cada número se reemplaza por el mínimo común múltiplo de los dos números vecinos a él. Si se obtiene el mismo número n en cada uno de los siete puntos, determine el menor valor posible para n.[/url]

Given two non-parallel segments $AB$ and $CD$ on the plane, find the locus of points $P$ on the plane such that the area of triangle $ABP$ is equal to the area of triangle $CDP$.

Let $\triangle{ABC}$ a triangle with $\angle{CAB}=90^{\circ}$ and $L$ a point on the segment $BC$. The circumcircle of triangle $\triangle{ABL}$ intersects $AC$ at $M$ and the circumcircle of triangle $\triangle{CAL}$ intersects $AB$ at $N$. Show that $L$, $M$ and $N$ are collinear.

In triangle $ABC$, points $P$, $Q$, and $R$ are marked on the sides $AB$, $BC$, and $AC$ respectively. The lengths of the sides of triangle $PQR$ are known to be 7, 8, and 9 centimeters. Find the radii of the circles inscribed in triangles $APR$, $BPQ$, and $CQR$ given that all three circles are tangent to the incircle of triangle $PQR$. [i]Proposed by Giorgi Arabidze, Georgia[/i]

Determine all the numbers formed by three different and non-zero digits, such that the six numbers obtained by permuting these digits leaves the same remainder after the division by $4$.

Prove that one can find infinite number of distinct pairs of integers such that every digit of each number is no less than $7$ and the product of two numbers in each pair is also a number with all its digits being no less than $7$. (6)

Find all real numbers $a$ for which there exist functions $f,g: \mathbb{R} \to \mathbb{R}$, where $g$ is strictly increasing, such that $f(1) = 1$, $f(2) = a$, and \[ f(x) - f(y) \leq (x-y)(g(x) - g(y)) \] for all real numbers $x$ and $y$.

(1) A function is defined $ f(x) \equal{} \ln (x \plus{} \sqrt {1 \plus{} x^2})$ for $ x\geq 0$. Find $ f'(x)$. (2) Find the arc length of the part $ 0\leq \theta \leq \pi$ for the curve defined by the polar equation: $ r \equal{} \theta\ (\theta \geq 0)$. Remark: [color=blue]You may not directly use the integral formula of[/color] $ \frac {1}{\sqrt {1 \plus{} x^2}},\ \sqrt{1 \plus{} x^2}$ here.

Let $(G,\cdot)$ be a finite group with the identity element, $e$. The smallest positive integer $n$ with the property that $x^{n}= e$, for all $x \in G$, is called the [i]exponent[/i] of $G$. (a) For all primes $p \geq 3$, prove that the multiplicative group $\mathcal G_{p}$ of the matrices of the form $\begin{pmatrix}\hat 1 & \hat a & \hat b \\ \hat 0 & \hat 1 & \hat c \\ \hat 0 & \hat 0 & \hat 1 \end{pmatrix}$, with $\hat a, \hat b, \hat c \in \mathbb Z \slash p \mathbb Z$, is not commutative and has [i]exponent[/i] $p$. (b) Prove that if $\left( G, \circ \right)$ and $\left( H, \bullet \right)$ are finite groups of [i]exponents[/i] $m$ and $n$, respectively, then the group $\left( G \times H, \odot \right)$ with the operation given by $(g,h) \odot \left( g^\prime, h^\prime \right) = \left( g \circ g^\prime, h \bullet h^\prime \right)$, for all $\left( g,h \right), \, \left( g^\prime, h^\prime \right) \in G \times H$, has the [i]exponent[/i] equal to $\textrm{lcm}(m,n)$. (c) Prove that any $n \geq 3$ is the [i]exponent[/i] of a finite, non-commutative group. [i]Ion Savu[/i]

Find all functions $f: \mathbb{R} \to \mathbb{R}$ such that \[f(x)f(y) = (x+y+1)^2 \cdot f \left( \frac{xy-1}{x+y+1} \right)\] $\forall x,y \in \mathbb{R}$ with $x+y+1 \neq 0$ and $f(x) > 1$ $\forall x > 0.$

In the corners of triangle $ABC$ there are three circles with the same radius. Each of them is tangent to two of the triangle's sides. The vertices of triangle $MNK$ lie on different sides of triangle $ABC$, and each edge of $MNK$ is also tangent to one of the three circles. Likewise, the vertices of triangle $PQR$ lie on different sides of triangle $ABC$, and each edge of $PQR$ is also tangent to one of the three circles (see picture below). Prove that triangles $MNK,PQR$ have the same inradius. [img]https://i.imgur.com/bYuBabS.png[/img]